Ultrametric Cantor sets and growth of measure

Abstract

A class of ultrametric Cantor sets (C, d u ) introduced recently (Raut, S and Datta, D P (2009), Fractals, 17, 45-52) is shown to enjoy some novel properties. The ultrametric d u is defined using the concept of relative infinitesimals and an inversion rule. The associated (infinitesimal) valuation which turns out to be both scale and reparametrisation invariant, is identified with the Cantor function associated with a Cantor setC where the relative infinitesimals are supposed to live in. These ultrametrics are both metrically as well as topologically inequivalent compared to the topology induced by the usual metric. Every point of the original Cantor set C is identified with the closure of the set of gaps ofC. The increments on such an ultrametric space is accomplished by following the inversion rule. As a consequence, Cantor functions are reinterpreted as locally constant functions on these extended ultrametric spaces. An interesting phenomenon, called growth of measure, is studied on such an ultrametric space. Using the reparametrisation invariance of the valuation it is shown how the scale factors of a Lebesgue measure zero Cantor set might get deformed leading to a deformed Cantor set with a positive measure. The definition of a new valuated exponent is introduced which is shown to yield the fatness exponent in the case of a positive measure (fat) Cantor set. However, the valuated exponent can also be used to distinguish Cantor sets with identical Hausdorff dimension and thickness. A class of Cantor sets with Hausdorff dimension log 3 2 and thickness 1 are constructed explicitly.

Nonlinear Scale invariant Formalism and its Application to Some Differential Equations

“…This is unlike the ordinary analysis, when one interprets 0 as a connected subset of R , thereby forcing v to vanish uniquely, so as to recover the usual structure of R . The above vanishing derivative can be interpreted nontrivially as a LCF [10] when R x  is supposed to belong to a Cantor subset of I .…”

“…Nontrivial inversion induced variations are revealed only under double logarithmic scales of an ordianry linear variable, when there is a transition from one gap to another (that is to say, between two points of the underlying Cantor set , for more details see [9,10]). So far we have discussed about the scale free analysis.…”

Excitation of flow instabilities due to nonlinear scale invariance

“…Here, we present in brief the mathematical arguments [17,19,20] leading to the emergence of multifractal scalings from standard differential measures in a laminar flow as the original laminar flow tends to become turbulent. Recall that the traditional (differential) Lebesgue measure is well suited for simple systems, for instance, the uniform rolling of a billiard ball along a straight line, say, or in a laminar flow.…”

“…Finally, to justify the anomalous scaling for t as t → ∞ in the present formalism let us remark that as η = t −1 → 0 respecting 0 <η n < n < t −1 , relatively invisible smaller scalesη n residing in (0, n ) might have a coherent, cooperative effect on the visible variable η in the form η −α( ) where the slowly varying, locally constant effective exponent α( ) = lim n→∞ log −n ( n /η n ) > 0 is interpreted as an ultrametric valuation living in a multifractal set of microscopically small and macroscopically large scales [18,19,20]. Clearly, cascades of infinitesimally small scale invisible elementsη n are related dually (i.e.…”

“…Theory of fractional kinetics [11] using fractional calculus and fractional differential equations are considered by various authors [10,11,13,12] to offer a general theoretical framework to model anomalous scalings and transports. The present approach is not only independent of the fractional kinetics but is a natural extension of the classical analysis to accommodate multifractal scaling behaviours in a smooth (differentiable) manner ( [15] - [20]). We show, in particular, how the linear differential measures of the form dt or dx i in a laminar flow can be realized as smooth multifractal measures of the form dt α( ) and dx…”